Let \({\mathcal {G}}_{N}=(V_{N}, E_{N}), N\ge 2\) , be random geometric graphs
with \(N\)  vertices deployed on \([0, 1]^2\)  and an undirected edge between two vertices in \(V_N\)  existing if their physical
distance is not larger than
\(\sqrt{2/N}\)  [1]}, [2]}.
In this section, we consider finding eigenvectors associated with maximal eigenvalue 1 of
lowpass spline filters \({\bf H}_{0, m}^{\rm spln}=({\bf I}-{\bf L}^{\rm sym}/2)^m, m\ge 1\) ,
where \({\bf L}^{\rm sym}\)  is the symmetric normalized Laplacian matrix on
the graph \({\mathcal {G}}_N\) 
[1]}, [4]}. In the simulations, we take \(c=0.01\)  and use PGDA and PGDA1h to denote the
PGDA with \({\bf A}\)  replaced by \({\bf I}-{\bf H}_{0, m}^{\rm spln}\)  and \({\bf Q}\)  by \({\bf Q}_c\)  in (REF ) and \(\widehat{\bf Q}_c\) 
in (REF )
respectively, and similarly we use SPGDA and SPGDA1h to denote the SPGDA with \({\bf A}\)  replaced by \({\bf I}-{\bf H}_{0, m}^{\rm spln}\)  and \({\bf Q}\)  by
\({\bf Q}_c^{\rm sym}\)  in (REF ) and \(\widehat{\bf Q}_c^{\rm sym}\)  in (REF )
respectively. For the sequences
\({\bf x}_n, n\ge 0\) , in the PGDA, SPGDA, PGDA1h and SPGDA1h and their limits \({\bf u}\) , define
convergence errors
\({\rm CE}(n)={\rm log}_{10}\Vert \tilde{\bf x}_{n}-\tilde{\bf u} \Vert _2\) 
and normalized residues
\({\rm NR}(n)={\rm log}_{10}\Vert ({\bf I}-{\bf H}_{0, m}^{\rm spln}) \tilde{\bf x}_{n}\Vert _2, n\ge 0,\) 
in the logarithmic scale, where \(\tilde{\bf x}_{n}= {\bf x}_n /\Vert {\bf x}_n\Vert _2\) , \(\tilde{\bf u}={\bf u}/\Vert {\bf u}\Vert _2\) , and
\(\Vert {\bf x}\Vert _2=\big (\sum _{j\in V} |x(j)|^2\big )^{1/2}\)  for \({\bf x}=(x_j)_{j\in V}\) .
Shown in
Figure REF  are the average of convergence errors
\({\rm CE}(n)\) 
and normalized residues
\({\rm RE}(n), n\ge 0,\) 
over 500 trials. This demonstrates the exponential convergence
of the sequence \({\bf x}_n, n\ge 0\) , in the proposed distributed iterative algorithms to eigenvectors
associated with eigenvalue 1 of lowpass spline filters, which is proved in
Theorems REF  and REF .
